So if I define a sequence that converges to a point at 0, f would be able to able to map that into the co-domain, would that be the right start?

Yes, that is a good way to start. Now prove that the sequence is Cauchy, and hence has a limit. Then you'll need to check that this limit is the same for all sequences .

(A quicker way to say the same thing is that a function with a bounded derivative is uniformly continuous; and if a function is uniformly continuous on a set A then it extends continuously to the closure of A.)

Please look at Prof. Opalg’s second comment about uniform continuity.
A function with a bounded derivative is uniformly continuous.
Now go back to the last question I helped you with.
I think that was a lemma for this question.